The actin cortex is a complex cytoskeletal machinery which drives and responds to changes in cell shape. It must generate or adapt to plasma membrane curvature to facilitate diverse functions such as cell division, migration and phagocytosis. Due to the complex molecular makeup of the actin cortex, it remains unclear whether actin networks are inherently able to sense and generate membrane curvature, or whether they rely on their diverse binding partners to accomplish this. Here, we show that curvature sensing and generation is an inherent capability of branched actin networks nucleated by Arp2/3 and VCA. We develop a robust method to encapsulate actin inside giant unilamellar vesicles (GUVs) and assemble an actin cortex at the inner surface of the GUV membrane. We show that actin forms a uniform and thin cortical layer when present at high concentration and distinct patches that generate negative membrane curvature at low concentration. Serendipitously, we find that the GUV production method also produces dumbbell-shaped GUVs, which we explain using mathematical modelling in terms of membrane hemifusion of nested GUVs. We find that dendritic actin networks preferentially assemble at the neck of the dumbbells, which possess a micron-range convex curvature that matches the curvature generated by actin patches in spherical GUVs. Minimal dendritic actin networks can thus both generate and sense membrane curvatures, which may help mammalian cells to robustly recruit actin to curved membranes in order to facilitate diverse cellular functions such as cytokinesis and migration.